The AB label is regulated yes, which is almost equivalent to the EU green leaf. Then there are various private labels. In the US it's all up to private labels I believe.

Anyone can put "bio" and a vaguely green packaging on anything though AFAIK. And I don't think the average consumer is very knowledgeable about which label means what; I certainly am not.

Then there's the problem of fraud, and various issues with the way the EU defines "biological agriculture", but I don't really know much about either.